Abel's theorem in problems and solutions - School of Mathematics

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244 Appendix by Khovanskii Let us analyze the case in which the group H(G) is finite and soluble. This happens if and only if the polygon G is sent by a homographic transformation to a polygon whose sides lie on a net of great circles different from that of the dodecahedron–icosahedron. In this case the group L(G) is soluble, and the function in expressed in terms of rational functions by means of arithmetic operations and of radicals (cf., §A.8). From our results a theorem follows: THEOREM ON THE POLYGONS BOUNDED BY OF ARCS OF CIRCLES ([6],[8],[10]). For an arbitrary polygon G not belonging to the three cases of integrability above, the function not only is not representable by generalized quadratures, but it cannot be expressed in terms of singlevalued by means of generalized quadratures, compositions, and meromorphic operations. A.11 Topological obstructions for the solvability of differential equations A.11.1 The monodromy group of a linear differential equation and its relation with the Galois group Consider the linear differential equation where the are rational functions of the complex variable The poles of the functions and are called the singular points of the equation (A.14). Near a non-singular point the solutions of the equations form a space of dimension Consider now an arbitrary curve on the complex plane, beginning at and ending at the point and avoiding the singular points The solutions of the equation can be analytically continued along the curve, remaining solutions of the equation. Hence to every curve there corresponds a linear mapping of the space of the solutions at the point in the space of the solutions at the point If one changes the curve avoiding the singular points and leaving its ends fixed, the mapping does not vary. Hence to a closed curve there
Solvability of Equations 245 corresponds a linear transform of the space into itself. The totality of these linear transforms of the space forms a group which is called the monodromy group of the equation (A. 14). So the monodromy group of an equation is the group of the linear transforms of the solutions which correspond to different turns around the singular points. The monodromy group of an equation characterizes the property of its solutions being multi- valued. Near a non-singular point there are linearly independent solutions, of the equation (A.14). In this neighbourhood one can consider the field of functions that is obtained by adding to the field of rational functions all solutions and all their derivatives. Every transformation of the monodromy group of the space of solutions can be continued to an automorphism of the entire field Indeed, with functions along the curve every element of the field can be analytically continued. This continuation gives the required automorphism, because during the continuation the arithmetic operations and the differentiation are preserved, and the rational functions come back to their initial values because of their uniqueness. In this way the monodromy group of an equation is contained in its Galois group. The field of the invariants of the monodromy group is a subfield of consisting of the single-valued functions. Differently from the algebraic case, for differential equations the field of invariants under the action of the monodromy group can be bigger than the field of rational functions. For example, for the differential equation (A.14), in which all the coefficients are polynomials, all solutions are single-valued. But of course the solutions of such equations are not always polynomials. The reason is that here the solutions of differential equations may grow exponentially in approaching the singular points. One knows an extension of the class of linear differential equations for which there are no similar complications, i.e., for which the solutions, whilst approaching the singular points, grow at most as some power. Differential equations which possess this property are called equations of Fuchs’ type. For differential equations of Fuchs’ type the Frobenius theorem holds. THEOREM 1. For the differential equations of Fuchs’ type the subfield of the differential field that consists of single-valued functions coincides with the field of rational functions. According to the differential Galois theory, from the Frobenius the-
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ABEL’S THEOREM IN PROBLEMS AND SO
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Abel’s Theorem in Problems and So
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Contents Preface for the English ed
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A.10.3 The integrable case A.11 Top
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Preface to the English edition by V
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of the differential equations. Unfo
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Preface In high school algebraic eq
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Introduction We begin this book by
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Introduction 3 where by is indicate
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Introduction 5 By Viète’s theore
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Introduction 7 We will be able to p
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Chapter 1 Groups 1.1 Examples In ar
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Groups 11 the triangle ABC into its
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Groups 13 7. Find all symmetries of
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Groups 15 notations (see the soluti
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Groups 17 element, which we will de
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Groups 19 35. Suppose that is the m
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Groups 21 If a group is isomorphic
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Groups 23 tude); (rotation by 180°
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Groups 25 Hence left cosets generat
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Groups 27 as a permutation of the t
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Groups 29 102. Let be the order of
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Groups 31 111. Find all normal subg
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Groups 33 FIGURE 8 127. Prove that
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Groups 35 137. Prove that ker is a
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Groups 37 We now observe what happe
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Groups 39 FIGURE 12 vertices; 4) ro
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Groups 41 Every permutation of degr
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Groups 43 180. Prove that by multip
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Chapter 2 The complex numbers When
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The complex numbers 47 195. Prove t
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The complex numbers 49 where is the
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The complex numbers 51 Consequently
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The complex numbers 53 For complex
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The complex numbers 55 think that n
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The complex numbers 57 214. Let us
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The complex numbers 59 219. Prove t
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The complex numbers 61 functions (s
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The complex numbers 63 of continuit
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The complex numbers 65 erations of
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The complex numbers 67 a) b) c) d)
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The complex numbers 69 REMARK. If a
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The complex numbers 71 2.8 Images o
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The complex numbers 73 where all th
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The complex numbers 75 in such func
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The complex numbers 77 FIGURE 26 no
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The complex numbers 79 C from to th
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The complex numbers 81 DEFINITION.
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The complex numbers 83 FIGURE 32 cu
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The complex numbers 85 to infinity
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The complex numbers 87 property is
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The complex numbers 89 means will b
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The complex numbers 91 For example,
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The complex numbers 93 of the Riema
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The complex numbers 95 in Figure 38
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The complex numbers 97 permutation
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The complex numbers 99 2.13 Monodro
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The complex numbers 101 First we pr
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The complex numbers 103 and prove t
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Chapter 3 Hints, Solutions, and Ans
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Solutions 107 positive integer unde
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Solutions 109 on the left and by on
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Solutions 111 39. a) See Table 8; b
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Solutions 113 of the real numbers i
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Solutions 115 to all and therefore
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Solutions 117 is a subgroup of the
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Solutions 119 hypothesis. The group
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Solutions 121 The given group is th
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Solutions 123 of symmetries of the
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Solutions 125 i.e., there exists an
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Solutions 127 Answer. The normal su
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Solutions 129 In this way the quoti
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Solutions 131 sends every into itse
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Solutions 133 ement of the commutan
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Solutions 135 143. Let be the three
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Solutions 137 150. See 57. Suppose
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Solutions 139 subgroup. Hence if a
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Solutions 141 is commutative. Since
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Solutions 143 180. Since the lower
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Solutions 145 FIGURE 43 190. The pe
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Solutions 147 permutations of type
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Solutions 149 is a field. 195. We h
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Solutions 151 necessarily that and
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Solutions 153 Let C be the field of
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Solutions 155 in as numerators and
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Solutions 157 c) d) e) where 226. W
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Solutions 159 function of real argu
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Solutions 161 i.e., Consider the re
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Solutions 163 Choose as the smalles
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Solutions 165 Since the function is
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Solutions 167 FIGURE 55 FIGURE 56 F
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Solutions 169 FIGURE 60 257. Suppos
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Solutions 171 265. 266. 267. If the
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Solutions 173 following way. If the
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Solutions 175 for all such that and
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Solutions 177 b) In order for to va
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Solutions 179 to the condition coin
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Solutions 181 are continuous functi
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Solutions 183 turn around the other
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Solutions 185 sought is shown in Fi
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Solutions 187 FIGURE 86 FIGURE 87 p
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Solutions 189 sum of multi-valued f
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Solutions 191 316. a) Let and be th
Page 209 and 210: Solutions 193 by the formal method,
Page 211 and 212: Solutions 195 The correct scheme is
Page 213 and 214: Solutions 197 branch of the functio
Page 215 and 216: Solutions FIGURE 110 FIGURE 111 329
Page 217 and 218: Solutions 201 It follows that and b
Page 219 and 220: Solutions 203 340. To every sheet o
Page 221 and 222: Solutions 205 Since by hypothesis t
Page 223 and 224: Solutions 207 moving along a curve
Page 225 and 226: Drawings of Riemann surfaces 209 Th
Page 227 and 228: FIGURE 119 211
Page 237 and 238: Appendix by A. Khovanskii: Solvabil
Page 239 and 240: Solvability of Equations 223 functi
Page 241 and 242: Solvability of Equations 225 Suppos
Page 243 and 244: Solvability of Equations 227 where
Page 245 and 246: Solvability of Equations 229 finds
Page 247 and 248: Solvability of Equations 231 quadra
Page 249 and 250: Solvability of Equations 233 More p
Page 251 and 252: Solvability of Equations 235 group.
Page 253 and 254: Solvability of Equations 237 Every
Page 255 and 256: Solvability of Equations 239 The gr
Page 257 and 258: Solvability of Equations 241 FIG. 1
Page 259: Solvability of Equations 243 are ob
Page 265 and 266: Solvability of Equations 249 up to
Page 267 and 268: Solvability of Equations 251 fined
Page 269 and 270: Solvability of Equations 253 the qu
Page 271 and 272: Solvability of Equations 255 the no
Page 273 and 274: Solvability of Equations 257 THEORE
Page 277 and 278: Bibliography [1] [2] [3] [4] [5] [6
Page 279 and 280: [23] [24] [25] [26] [27] [28] 263 V
Page 281 and 282: Appendix (V.I. Arnold) The topologi
Page 283 and 284: Index Abel’s theorem, 6, 103 addi
Page 285: pack of sheets, 96 parametric equat
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Abel's theorem in problems and solutions - School of Mathematics

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